In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 16x

Verified step by step guidance

1. insert step 1> Factor out the greatest common factor (GCF) from the polynomial. In this case, the GCF is \(x\).
2. insert step 2> After factoring out \(x\), the expression becomes \(x(x^2 - 16)\).
3. insert step 3> Recognize that \(x^2 - 16\) is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).
4. insert step 4> Apply the difference of squares formula to \(x^2 - 16\), where \(a = x\) and \(b = 4\), resulting in \((x - 4)(x + 4)\).
5. insert step 5> Combine the factored terms to express the completely factored form of the polynomial: \(x(x - 4)(x + 4)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial expression into simpler components, or factors, that when multiplied together yield the original polynomial. This process is essential for solving polynomial equations and simplifying expressions. Common techniques include identifying common factors, using special products, and applying methods like grouping.

Difference of Squares

The difference of squares is a specific factoring pattern that applies to expressions of the form a² - b², which can be factored into (a - b)(a + b). In the given polynomial x³ - 16x, recognizing that 16x can be expressed as (4√x)² allows us to apply this pattern effectively, simplifying the factoring process.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with real coefficients. Understanding whether a polynomial is prime is crucial for determining its factorability. In the context of the given polynomial, recognizing its structure helps in identifying whether it can be factored further or if it remains irreducible.